Cardinal invariants of infinite groups

LetG be a group. CallG akC-group if every element ofG has less thank conjugates. Denote byP(G) the least cardinalk such that any subset ofG of sizek contains two elements which commute.It is shown that the existence of groupsG such thatP(G) is a singular cardinal is consistent withZFC. So is the existence of groupsG which are notkC but haveP(G) wherek is a limit cardinal. On the other hand, ifk is a singular strong limit cardinal, andG is akC-group, thenP(G)k. This partially answers questions, and improves results, of Faber, Laver and McKenzie.The present paper has non-trivial intersection with the author's Diplomarbeit written under the direction of Prof. Ulrich Felgner at the University of Tübingen, W. Germany, 1988     

The computations of some Schur indices

Let χ be an irreducible character of a finite groupG. Letp=∞ or a prime. Letm _p (χ) denote the Schur index of χ overQ _p , the completion ofQ atp. It is shown that ifx is ap′-element ofG such that for all irreducible charactersX _u ofG thenm _p (χ)/vbχ(x). This result provides an effective tool in computing Schur indices of characters ofG from a knowledge of the character table ofG. For instance, one can read off Benard’s Theorem which states that every irreducible character of the Weyl groupsW(E _n), n=6,7,8 is afforded by a rational representation. Several other applications are given including a complete list of all local Schur indices of all irreducible characters of all sporadic simple groups and their covering groups (there is still an open question concerning one character of the double cover of Suz). This work was partly supported by NSF Grant MCS-8201333.     

The construction of affine structures on virtually nilpotent groups

In this paper we establish faithful affine representations for 2-step nilpotent Lie groupsG and the associated groupsG×Aut (G), which play a crucial role in the theory of infra-nilmanifolds. Afterwards, we apply the obtained results, in order to find representations of 3-step Almost-crystallographic groups. Finally, we indicate how all of this might be used to compute the second cohomology group of an Almost-crystallographic group with coefficients in a free abelian group.     

On simple groups with a cyclic maximal 2-Sylow intersection

Finite simple groupsG with a cyclic maximal 2-Sylow intersectionV are classified under the assumption that [G: N _G (V)] is odd.     

Clifford classes for isoclinic groups

Let G be a finite group, N a normal subgroup of G, and an irreducible character of G. Clifford Theory studies a whole collection of related irreducible characters of all the subgroups of G that contain N. The relationships among these characters as well as their Schur indices are controlled by the Clifford class c Clif(G/N, F) of with respect to N over some field F. This is an equivalence class of central simple G/N-algebras. Assume now that G/N is cyclic. One can obtain a new isoclinic group and character by multiplying each element of each coset of N in G by an appropriate power of a fixed root of unity . We show that there is a simple formula to calculate the Clifford class of in terms of c and . Hence, the Clifford class c controls not only the Schur index of the characters of all the subgroups of G that contain N, it also controls the Schur indices of the characters of the corresponding characters of the isoclinic groups When is a |G/N|-th root of 1, our formula shows that then When = i and |G/N| = 2, the implicit transformation on Clif(Z/2Z, F) yields a group homomorphism of the group structure introduced on the Brauer-Wall group of F to describe the Schur indices of all the irreducible characters of the double covers of the symmetric and alternating groups.Received: 17 August 2001     

Non-commutative spaces with transitive translation groups

Let G be a finite group. The problem of finding all strongly skewaffine spaces with transitive translation group isomorphic to G will be reduced to the determination of all Schur rings over G.Dedicated to Helmut Salzmann on his 60th birthday     

On simple groups with a quaternion maximal 2-sylow intersection

Finite simple groupsG with a generalized quaternion maximal 2-Sylow intersectionV are determined under the assumption that [G: N _G (V)] is odd.     

A characterization of a class of [Z] groups via Korovkin theory

We characterize all the central topological groupsG for which the centreZ(L ¹(G)) of the group algebra admits a finite universal Korovkin set. It is proved thatZ(L ¹(G)) has a finite universal Korovkin set iffĜ is a finite dimensional, separable metric space. This is equivalent to the fact thatG is separable, metrizable andG/K has finite torsion free rank, whereK is a compact open normal subgroup of certain direct summand ofG.     

Finite groups with many product conjugacy classes

We classify all finite groupsG such that the product of any two non-inverse conjugacy classes ofG is always a conjugacy class ofG. We also classify all finite groupsG for which the product of any twoG-conjugacy classes which are not inverse modulo the center ofG is again a conjugacy class ofG.     

Symmetry groups of vertex-transitive polytopes

Groups which are not isomorphic to the symmetry group of any vertextransitive polytope (of any dimension) are characterized as generalized dicyclic, or abelian groups but not elementary 2-groups. The same class of groupsG is also characterized by the existence of a permutation groupP acting onG, containingG* (the regular representation ofG) as a proper subgroup, such that the members of the stabilizerP _u of the unitu ε G take everyg ε G tog ^±1.     

THE ACTION OF THE LINEAR CHARACTER GROUP ON THE SET OF NON-LINEAR IRREDUCIBLE CHARACTERS

Let G be a nonabelian finite group. Then Irr(G/G′) is an abelian group under the multiplication of characters and acts on the set of non-linear irreducible characters of G via the multiplication of characters. The purpose of this paper is to establish some facts about the action of linear character group on non-linear irreducible characters and determine the structures of groups G for which either all the orbit kernels are trivial or the number of orbits is at most two. Using the established results on this action, it is very easy to classify groups G having at most three nomlinear irreducible characters.     

Limits of commutative triangular systems on simply connected step 2-nilpotent Lie groups

We prove that on simply connected step 2-nilpotent Lie groupsG any limit of a commutative infinitesimals triangular system of probability measures which are either all symmetric or supported by some discrete subgroupH is infinitely divisible onG resp.H.     

The Schur property for subgroup lattices of groups

A classical theorem of Schur states that if the centre of a group G has finite index, then the commutator subgroup G′ of G is finite. A lattice analogue of this result is proved in this paper: if a group G contains a modularly embedded subgroup of finite index, then there exists a finite normal subgroup N of G such that G/N has modular subgroup lattice. Here a subgroup M of a group G is said to be modularly embedded in G if the lattice is modular for each element x of G. Some consequences of this theorem are also obtained; in particular, the behaviour of groups covered by finitely many subgroups with modular subgroup lattice is described. Received: 16 October 2007, Final version received: 22 February 2008     

Regular and normal closure operators and categorical compactness for groups

For a class of groupsF, closed under formation of subgroups and products, we call a subgroupA of a groupG F-regular provided there are two homomorphismsf, g: G » F, withF F, so thatA = {x G |f(x) =g(x)}.A is calledF-normal providedA is normal inG andG/A F. For an arbitrary subgroupA ofG, theF-regular (respectively,F-normal) closure ofA inG is the intersection of allF-regular (respectively,F-normal) subgroups ofG containingA. This process gives rise to two well behaved idempotent closure operators.A groupG is calledF-regular (respectively,F-normal) compact provided for every groupH, andF-regular (respectively,F-normal) subgroupA ofG × H, ₂(A) is anF-regular (respectively,F-normal) subgroup ofH. This generalizes the well known Kuratowski-Mrówka theorem for topological compactness.In this paper, theF-regular compact andF-normal compact groups are characterized for the classesF consisting of: all torsion-free groups, allR-groups, and all torsion-free abelian groups. In doing so, new classes of groups having nice properties are introduced about which little is known.     

Complete embeddings of linear orderings and embeddings of lattice-ordered groups

An infinite linearly ordered set (S,≦) is called doubly homogeneous, if its automorphism group Aut(S,≦) acts 2-transitively on it. We study embeddings of linearly ordered sets into Dedekind-completions of doubly homogeneous chains which preserve all suprema and infima, and obtain necessary and sufficient conditions for the existence of such embeddings. As one of several consequences, for each lattice-ordered groupG and each regular uncountable cardinalκ≧|G | there are 2⋉ non-isomorphic simple divisible lattice-ordered groupsH of cardinalityκ all containingG as anl-subgroup.     

Generalized Frobenius groups

A pair (G. K) in whichG is a finite group andK◃G, 1<K<G, is said to satisfy (F2) if |C _G (x)|=|C _G/K (xK)| for allx∈G/K. First we survey all the examples known to us of such pairs in whichG is neither ap-group nor a Frobenius group with Frobenius kernelK. Then we show that under certain restrictions there are, essentially, all the possible examples.     

On (2,n)-groups related to fibonacci groups

In this paper, we study certain groupsG generated by two elementsa andb of orders 2 andn respectively subject to one further defining relation, and determine their structure. We also point out certain connections between these groups and the Fibonacci groupsF(r, n).     

Group rings of hyperabelian groups

Ifk is a field of characteristicp>0 then we find all hyper-(Abelian or locally finite-p′) groupsG such that the augmentation ideal of the group algebrakG has the AR property. The second author is partially supported by NSF grant MCS-7828082.     

On the computation of the projective surgery obstruction groups

The computation of the projective surgery obstruction groupsL _n ^p (ZG), forG a hyperelementary finite group, is reduced to standard calculations in number theory, mostly involving class groups. Both the exponent of the torsion subgroup and the precise divisibility of the signatures are determined. ForG a 2-hyperelementary group, theL _n ^p (ZG) are detected by restriction to certain subquotients ofG, and a complete set of invariants is given for oriented surgery obstructions.Partially supported by NSERC grant A4000.Partially supported by NSF grant DMS-8610730(1) and the Danish Research Council.     

Straightening and bounded cohomology of hyperbolic groups

It was stated by M. Gromov [Gr2] that, for any hyperbolic group G, the map from bounded cohomology Hⁿ_b(G,\Bbb R) H^n_b(G,{\Bbb R}) to Hⁿ(G,\Bbb R) H^n(G,{\Bbb R}) induced by inclusion is surjective for n 3 2 n \ge 2 . We introduce a homological analogue of straightening simplices, which works for any hyperbolic group. This implies that the map Hⁿ_b(G,V) ? Hⁿ(G,V) H^n_b(G,V) \to H^n(G,V) is surjective for n 3 2 n \ge 2 when V is any bounded \Bbb QG {\Bbb Q}G -module and when V is any finitely generated abelian group.